Nexus 2002, 15-18 June 2002, Óbidos, Portugal


João Pedro Xavier
Av. Fernão Magalhães, 420 - 1º Esq.
Douro Litoral 4300-189 PORTUGAL

This paper is a first approach to Mestre António Rodrigues' masterpiece, The Onze Mil Virgens Chapel (c. 1565) at Alcácer do Sal, a classic temple built as a mausoleum for the cosmopolitan noble Dom Pedro de Mascarenhas. As in his other works, namely Santa Maria da Graça Church at Setúbal (c. 1570), here we are able to disclose in several ways the omnipresence of geometry, always in association with number and calculus, being this condition essential to the definition of an idea of architecture with a strong mathematical support.

N2002-XavierLeft, Chapel elements and their ad quadratum geometry;
Right, corner of the chapel showing the pendentives and blind oculi

Although this temple is tied to a pre-existing one, the Church of the Convent of Santo António, it has a rigorous geometric plan based on the square, a figure we can detect at different scales. An ad quadratum geometry is almost always present in the definition of the general proportions of the Chapel, as the side of the squared modular piers and pilasters and its diagonal are related with the measures of the whole, and it is clearly stated at the main space of the building, the sepulchral chapel, almost a cube crowned with an amazing translucent cosmic dome in pink marble from Estremoz.

But we have also in this temple the frequent use of the 5:4 rectangle, "a sequisquartal proposition of a square and a quarter" as Rodrigues defines it himself in his incomplete Treatise of Architecture (1576-1579). We can find it in the making of different spaces (the plan of the main chapel and sacristy, the transversal section of the nave) and in some architectural elements (the altar window and the blind niches between the serliana) and verify that they are part of a geometrical progression whose ratio is 7:6, the same form of the back wall of the reliquary where a exquisite statue of Christ stands.

I suspect that the use of the 5:4 rectangle could be another cosmic sign as it is related to the height of the celestial pole at Alcácer. Following Vitruvius, if we measure there a gnomon according to the larger side of that rectangle, the length of its noonday equinoctial shadow will be its lesser side. I think there is a strong possibility that this relationship is intentional attending to the references concerning these matters in his Treatise and also because of his connection with the Great Cosmographer of the Realm, Pedro Nunes (born at Alcácer).

I also used geometry with the intrinsic stylistic coherence of the composition and some material evidences to make hypothesis for a possible design of the main façade which remains one of the most puzzling aspects of this temple as it has been adulterated, or may even never been built, due to the addition of a two-storied body that sits transversally to the Chapel and the Santo António Church. Nevertheless we still have the main portico, whose measures depend on the temple module - the square pillar, clearly related to the one presented in Serlio's Treatise and redesigned by António Rodrigues in his own.

Mainly with this work, his solid theoretical formation and its pedagogical side, António Rodrigues should be considered one of the most important Portuguese architects of his time… with a scientific inclination, evidently!

This email address is being protected from spambots. You need JavaScript enabled to view it.received his degree in Architecture from the Faculty of Architecture of the University of Porto (FAUP) and is licensed as an architect at the College of Architects in Porto since 1986. He won the following scholar prizes: "Prémio Florêncio de Carvalho" and "Prémio Engº António de Almeida".
He worked in Álvaro Siza's office from 1986 to 1999. At the same time he set his own practice as an architect. He has participated in several exhibitions, courses and seminaries. One of his latest projects was the Exhibition "Matemática Viva" (an interactive exhibition on mathematics), at the Pavilhão do Conhecimento in Lisbon, organized by the Association ATRACTOR, where he conceived also all the modules on perspective.
He has been teaching Geometry since 1985: at Architecture School of Cooperativa Árvore in Porto, Fine Arts School of Porto and at FAUP from 1991 onwards. At 1996 he made the work Perspectiva, perspectiva acelerada e contraperspectiva, published by FAUP Publicações at 1997, and became assistant lecturer of that Chair. Now he is preparing his Phd on the same subject, advised by Prof. Arch. Alexandre Alves Costa.
Xavier has always been interested in the relationship between architecture and mathematics, especially geometry. He published several works and papers on the subject, made conferences and lectures and gave courses to high school teachers. He also collaborated with the Ministry of Education coordinating the team in charge of the elaboration of Descriptive Geometry curricula in Portugal.

The correct citation for this article is:
João Pedro Xavier, "António Rodrigues, a Portuguese architect with a scientific inclination", pp. 253-268 in Nexus IV: Architecture and Mathematics, eds. Kim Williams and Jose Francisco Rodrigues, Fucecchio (Florence): Kim Williams Books, 2002.

Randy S. Swanson
History and Theory of Technology and Design
College of Architecture, UNC Charlotte
Charlotte NC. 28223-0001 USA

The Observatoire de Paris (1667-1672) by Claude Perrault, was originally intended to house the Paris Academy of Sciences established by Jean Baptiste Colbert for Louis XIV. The project is little known in architectural histories despite its importance as Perrault's only completed building design and as the first facility in the institutionalization of science. The work presents an understanding of physical geometry that rivals the application of geometry in astronomy during that era, suggesting a range of outside influences upon Perrault. On-site research in Paris involving both field and archival efforts took place during the summers of 1997 and 1999. These examinations involved an instrumented confirmation of the building dimensions, vaulting geometry and structure, masonry construction, unique services and ornamentation, for concordance with existing original drawings.

Vault dimensions, rear stairwell, main floor, Observatoire de Paris (Randy Swanson, 1999)

A brief overview of the facility and the conceptual sources that influenced Perrault will be provided that will lead directly into the following two areas of development revealed by the on-site examination:

  1. A stereometric achievement - a cantilevered elliptical vaulted semi-helical stairwell form that suggests an advanced level of craft knowledge that is related to but challenges the theoretical efforts of Girard Desargues, (1591-1661) , or Abraham Bosse, in descriptive geometry until the era of Gaspard Monge (1746-1818).
  2. Subtle dimensional eccentricities that do not easily permit the work to be geometrically understood with the use of regular forms, nor allow the building to be easily resolved as a cube as Professor Perez-Gomez and others have previously suggested.

This paper will present evidence for a practical resolution of the complex stairwell form and compare it to the work of Desargues/Bosse. The paper will suggest that Perrault took the opportunity in the design of the Observatoire as a practical test of his own theories concerning architectural proportions as the justification for the geometrical eccentricities. The paper will close with a suggestion of the impact the Observatoire project in the formation of Perrault's views as presented in his Ordonnance for the Five Kinds of columns after the Method of the Ancients.

This email address is being protected from spambots. You need JavaScript enabled to view it. is a registered architect, an Associate Professor in the College of Architecture, and a member of the Graduate College at The University of North Carolina at Charlotte. He has a teaching responsibility in the area of the history and theory of technology and design. Prior to his academic career he was in practice. In Chicago, Illinois he worked in the firm of Keck and Keck, Architects, solar residential design pioneers. He also practiced in the Washington, D.C., area as the project architect for the rehabilitation and design of several scientific/military laboratories and medical facilities. His principal areas of research interest involve the history, evolution and design of technically complex facilities such as scientific laboratories and medical facilities. He is currently preparing a manuscript entitled The Place of Inquiry, which will present a history of the development of scientific space from 1660 to 1890. His current professional involvement includes the Society of Architectural Historians, Society for the History of Technology, Society of Industrial Archeology, and Society of Building Science Educators.

The correct citation for this article is:
Randy S. Swanson, "Geometry in Perrault's Observatoire", pp. 237-251 in Nexus IV: Architecture and Mathematics, eds. Kim Williams and Jose Francisco Rodrigues, Fucecchio (Florence): Kim Williams Books, 2002.

Peter Schneider
Department of Architecture
University of Colorado at Denver
Campus Box 126, P.O.Box 173364
Denver, CO 80217-3364 USA

There is a great deal of literature that revolves around the manipulation of the square through the geometrical processes of ad quadratum, the 'sacred cut,' and the 'golden rectangle'. They have been the topics of discussion at previous Nexus meetings. The processes that lead to the production of ad quadratum, the golden rectangle, and the sacred cut exploit the properties of the square, but they do so only by virtue of the existence of a primal figure, a first and original square, to operate on and make the subsequent exploitation of its properties possible. That 'first' square has to be the result of a simple, clear, effective and efficient method of construction: a method that constituted its original geometry and established its perfection as the instrument for all subsequent geometrical and metrological manipulations.

The paper deals directly with this puzzle of the first square, tracing its origins to the emergence of the western architectural tradition in ancient Egypt. The question that it will ask and answer is : "how did that original square - that first figure we manipulate in and through these geometrical processes - get there in the first place?" It will do this by showing that there are recurring references to a very specific pair of measures in the history of the architect's practices, and that there are recurring discussions of techniques for their application.

N2002-SchneiderStretching the Cord: An Egyptian geometer checking the right angle on a block of stone. Ippolito Rosellini, Monumenti Civile, Pisa, 1834.

Those references occur in the Roman period in the texts of Vitruvius and Faventius, in the early and late medieval texts by de Honnecourt, Roriczer, Schmuttermayer and Lechler, and in the Renaissance works of Alberti, Serlio and Palladio among others. Lorenz Lechler calls these measures the 'old' and the 'young' shoes, and makes the point that the young is always "taken out of the old." He also makes the point that these measures were established by the teachings of the 'old fathers,' and that his tradition held that they originate in ancient Egypt in the time of the first pharaohs.

As one looks at the history of measure in ancient Egypt, one finds references to a 'new' measure that is also taken out of the 'old.' It is called the remen, and it is derived directly from the measure called the great or royal cubit. Its numeric - and therefore mathematical - relationship to the cubit is identical to the close approximation of the 1:Ö2 relationship that Lechler - and de Honnecourt before him - established between his old and new shoes, and that Faventius and Vitruvius establish as the two measures essential to 'composing the square to perfect standards.'

The paper argues that the existence of two common measures in ancient Egypt - the 20 digit remen and the 28 digit cubit - essentially solves in an elegant and convincing way the puzzle of construction of the first square. It suggests that there was a simple, clear, effective and efficient technique using two cords, with one knotted in remen and the other knotted in cubits, that allowed the ancient Egyptian architect to construct that first square from which all subsequent squares and proportions were derived with an amazingly high degree of geometric accuracy. That technique - known to the Ancient Egyptians as 'the stretching of the cord' - had its origin in the time of the old kingdom in ancient Egypt: in the archaic time of Djoser, Imhotep, Snefru, Amenhotep and Kanofer, the 'old fathers,' and the time of the construction of the first great pyramids and the invention of the geometer's - and the architect's - art.

This email address is being protected from spambots. You need JavaScript enabled to view it.was born and educated in South Africa, where he taught and practiced architecture before moving to the United States in 1977. He has taught architecture and directed graduate and undergraduate architecture programs in Alabama, Louisiana and Colorado. He currently teaches architectural history, theory and design at the Boulder and Denver campuses of the University of Colorado. His scholarly and research interests have been focused on the history of the architect: on the architect's mind, methods and manners as these have occurred in history, and in the way that the interactions between these three forces have shaped the discipline's traditions. He, like Louis Kahn, loves beginnings, and so his particular focus has been on exploring and understanding the forces that shaped the origin of the architectural tradition, and on the persistent influence those original ideas have had in shaping and forming the persona of the architect. His writings on the history of the architect and the architect's methods and practices have been widely published, and he has lectured on the topic extensively. He is currently exploring and writing about the connections that exist between the myths of architecture's archaic origins and the traditions embedded in its first, archetypal artifacts.

The correct citation for this article is:
Peter Schneider, "The Puzzle of the First Square in Ancient Egyptian Architecture", pp. 207-221 in Nexus IV: Architecture and Mathematics, eds. Kim Williams and Jose Francisco Rodrigues, Fucecchio (Florence): Kim Williams Books, 2002.

Richard Padovan
28 Petersham Road
Richmond upon Thames
Surrey TW10 6UW UK

The plastic number, discovered by Dom Hans van der Laan (1904-91) in 1928 shortly after he had abandoned his architectural studies and become a novice monk, differs from all previous systems of architectural proportions in several fundamental ways. Its derivation from a cubic equation (rather than a quadratic one such as that which defines the golden section) is a response to the three-dimensionality of our world. It is truly aesthetic in the original Greek sense, i.e., its concern is not 'beauty' but clarity of perception. Its basic ratios, approximately 3:4 and 1:7, are determined by the lower and upper limits of our normal ability to perceive differences of size among three-dimensional objects. The lower limit is that at which things differ just enough to be of distinct types of size. The upper limit is that beyond which they differ too much to relate to each other; they then belong to different orders of size. According to Van der Laan, these limits are precisely definable. The mutual proportion of three-dimensional things first becomes perceptible when the largest dimension of one thing equals the sum of the two smaller dimensions of the other. This initial proportion determines in turn the limit beyond which things cease to have any perceptible mutual relation. Proportion plays a curcial role in generating architectonic space, which comes into being through the proportional relations of the solid forms that delimit it. Architectonic space might therefore be described as a proportion between proportions.

The order of size embraces seven consecutive types contained between eight measures

Born in 1935, Richard Padovan studied architecture at the Architectural Association, London (1952-57). Since then he has combined practice with teaching and writing on architecture. He believes, however, that his real architectural education began when in encountered the work and thought of the Dutch Benedictine architect Dom Hans van der Laan in 1974. His translation of Van der Laan's treatise Architectonic Space appeared in 1983, followed by a monograph, Dom Hans van der Laan, Modern Primitive, in 1994. In 1999 he published Proportion: Science, Philosophy, Architecture. His latest book, Towards Universality: Le Corbusier, Mies and De Stijl (2002), contrasts the grandiose philosophical ideals of European modernism with its failure to realize those aims, particularly in the building of cities.

The correct citation for this article is:
Richard Padovan, "Dom Hans Van Der Laan and the Plastic Number", pp. 181-193 in Nexus IV: Architecture and Mathematics, eds. Kim Williams and Jose Francisco Rodrigues, Fucecchio (Florence): Kim Williams Books, 2002.

Vini Nathan
Interior Design Program, School of Arch & Design
Philadelphia University
School House Lane & Henry Ave
Philadelphia PA 19144-5497 USA

N2002-NathanThe Vastu purusha mandala (Dhama, 1962)

During the medieval ages in India, no single dynastic power served as the undisputed dispenser of cultural and artistic ideas. However, despite their regional flourishes, Hindu temple designs displayed a remarkable unity of aesthetic purposes. This unified philosophy was codified into a system of rules or canons (a compendium of architectural guidelines) called the Vastushastras. These canons were the purview of the priestly class, were intentionally made very complex so that they were incomprehensible to even skilled building craftspeople and were seldom challenged.

Of all the canons and rules in the Vastushastras, the one that found the most favor with building designers from ancient times to the present day is the Vastu purusha mandala. The Vastu purusha mandala has been defined as "a collection of rules which attempt to facilitate the translation of theological concepts into architectural form." This law of proportions and rhythmic ordering of elements not only found full expression in temples, but extended to residential and urban planning as well. This paper argues that the influence of the Vastu purusha mandala extended beyond building activity to encompass the cultural milieu as well. The first section discusses the principles underlying the Vastu purusha mandala. The application of the Vastu purusha mandala in residential design and city planning is discussed in the second section. The implications of the mandalas on the social milieu are also identified. Finally, the current status of the mandala in contemporary Indian architecture and urban design are identified.

This email address is being protected from spambots. You need JavaScript enabled to view it. is the Director of and Associate Professor in Interior Design in the School of Architecture and Design at Philadelphia University, Philadelphia, USA. Her educational background includes degrees in architecture and interior design. She worked as an architect for Walker Group/CNI in New York, a full service design firm specializing in retail design. She has also taught in interior design and architecture programs at Virginia Tech, the University of Michigan, Eastern Michigan University, Michigan State University, the University of Minnesota, New York Institute of Technology and at her current institution, Philadelphia University. Her research interests include chaos theory, design processes, and research methods. She is currently researching the relationship between mathematics and architectural canons in traditional and contemporary Indian building designs.

The correct citation for this article is:
Vini Nathan, "Vastu Purusha Mandala", pp. 151-163 in Nexus IV: Architecture and Mathematics, eds. Kim Williams and Jose Francisco Rodrigues, Fucecchio (Florence): Kim Williams Books, 2002.

John G. Hatch
Associate Professor (Art History)
Dept. of Visual Arts
The University of Western Ontario
London, Ontario CANADA N6A 5B7

Italian architect Francesco Borromini (1599-1667) remains an enigma for scholars of seventeenth-century architecture. His buildings rarely follow the accepted design standards of the time and their meaning eludes the traditional readings established by his contemporaries. Added to this is the problem that Borromini was not forthcoming in divulging the rationale and sources for his architectural eclecticism. There are two reasons that account for this eclecticism: the rather unique personality of the man himself, and the fact that many of his sources of inspiration were far from traditional for an architect of the time. One such source, as this paper will present, is the astronomer Johannes Kepler (1571-1603) who is best known in the annals of the history of science for his three planetary laws of motion. Kepler forged both a methodology and interpretation of the universe that appealed to Borromini on many levels. The most important would be Kepler's belief in the divine geometry ruling the structure of the universe and the manner of interpreting it. Borromini's churches of S. Carlo alle Quattro Fontane (Rome: 1638-41) and S. Ivo della Sapienza (Rome: begun 1642) involve rather novel and complex geometric designs that are both the starting and endpoint in understanding these buildings. That the meaning of these designs is indebted partly to Kepler is further confirmed by Borromini's adoption at S. Carlo and S. Ivo of the astronomer's rather unique Trinitarian interpretation of the cosmos.


Nested sphere model of the solar system. Johannes Kepler, Mysterium Cosmographicum (1596)

This email address is being protected from spambots. You need JavaScript enabled to view it. is an associate professor at The University of Western Ontario, specializing in the history of modern art, and is Acting Chair of the Department of Visual Arts. His most recent work focuses on the influence of contemporary physics on modern art. Dr. Hatch's latest and upcoming
publications include an examination of the role of Machian epistemology in the paintings and theories of the Czech artist František Kupka, and two studies dealing with the impact of Relativity theory on the visual conceptualizations of space found in the work of the Russian artists El Lissitzky and Naum Gabo. Dr. Hatch has also written on Francis Bacon, Cindy Sherman, and Robert Rauschenberg, while undertaking a few excursions outside of the modern period into the area of geometry and architecture, specifically publishing an article on the possible rationale for the introduction of alexemata in archaic Greek temples. He is currently researching the role of relativistic theories in the architectural designs of Theo van Doesburg and Cornelius van Eesteren, and has also turned his attention recently to post-World War II art in Europe and, particularly, at a potential Baroque revival in Italian Art of 1945 to 1972.

The correct citation for this article is:
John G. Hatch, "The Science Behind Francesco Borromini's Divine Geometry", pp. 127-139 in Nexus IV: Architecture and Mathematics, eds. Kim Williams and Jose Francisco Rodrigues, Fucecchio (Florence): Kim Williams Books, 2002.

Steven Fleming
University of Newcastle
Callaghan NSW 2308 AUSTRALIA
School of Architecture and the Built Environment

Klaus-Peter Gast claims that a method known as plan analysis reveals hidden geometrical patterns in almost all of Kahn's buildings. On first appraisal, Gast's analysis would In his recently published book Louis I. Kahn: The Idea of Order (Birkhäuser, 1998), appear to confirm claims by various scholars that Kahn was a Platonist. Jencks, Burton, Scully, Brownlee, De Long, Auer, and Danto have compared Kahn's tendency to conceive buildings as instances of "forms" to Plato's doctrine that particular things participate in what Plato calls Forms or Ideas. According to Gast, Kahn goes one step further by secretly inscribing his buildings with a geometrical kind of Platonic Form, the square.

Using simple mathematics, this paper demonstrates that Gast's geometrical analyses of three of Kahn's buildings do not to tally with the dimensions indicated on Kahn's working drawings for those projects. These discrepancies prompt an inquiry into Kahn's approach to number and geometry which takes its direction from scholarship linking Kahn's "form and design" theory to Plato's theory of Forms.

Before proceeding, two points of qualification are made. First, there is nothing in Plato to suggest that buildings should bear hidden inscriptions. Second, Plato's theory of Forms does not suggest that buildings should participate in Forms other than those corresponding to their class names. In other words, particular houses participate in The House Itself, but they needn't also participate in The Square Itself.

N2002-FlemingThe treatment of the facade in the First Unitarian Church

In The Republic (524), Plato argues that the sensory perception of ones own fingers is sufficient to apprehend that we have five fingers on each hand. However, should we ask which of our fingers are large and which are small, given that these are relative terms, we're forced to use reason and consider the existence of the transcendent Forms, Largeness Itself and Smallness Itself. Hence the common sight of similar but non-identical units can be thought of as a challenge to empiricism, steering viewers towards Plato's rationalistic epistemology. As well as units, this rationale extends to plane and solid geometry as well, so that any ambiguity surrounding a building's shape may cause viewers to contemplate such Forms as The Square Itself and The Cube Itself.

On many levels, the visual apprehension of Kahn's Unitarian Church in Rochester causes mental distress and demands the operation of reason. Due to the radial distribution of light towers, viewers expect the auditorium to be square. Their discovery that it is not square after all, would, according to the logic of Plato's finger analogy, lead them to contemplate of The Square Itself. Similarly, glazing strips at The Kimbell Art Museum draw viewers' attention to the fact that the vaults are not elliptical as they might have at first expected. The façade treatment at Rochester also invites the use of reason, since the perception of any aspect, in the words of The Republic (524e), "seems to involve plurality as much as unity." Likewise, the scalelessness of Kahn's National Assembly in Dacca might cause viewers to ask if the buildings are large, and what is large.

This interpretation of number and geometry in Kahn's work may seem obscure, but it does provide an explanation of his intensions which is intrinsic to his own dualistic metaphysics. Kahn's buildings invite conflicting readings, reminding viewers not to trust their senses. As Kahn says, a great building evokes the "unmeasurable" realm of "form," just as the sight of fingers, according to Plato, evokes the purely intelligible realm of Forms.

This email address is being protected from spambots. You need JavaScript enabled to view it. is a lecturer in architectural history and theory at The University of Newcastle in Australia. As a PhD candidate, he is examining the Platonic nature of Louis Kahn's design philosophy. His research has involved a study trip to the Louis Kahn Collection in Philadelphia and an extensive study of Platonism as it pertains to architectural theory. Publications stemming from this work cover topics such as The Neoplatonic tradition in Western architecture, mimesis, "sacred" geometry, and the implications of Platonic philosophy to architecture.
Prior to commencing his doctoral studies, Steven had worked as an architect with the EJE Group in Newcastle, Australia, and with the Housing and Development Board (HDB) in Singapore, where he designed a 4.2 hectare neighbourhood park and a public housing project featuring 360 apartments in a sensitive military zone. He also managed three construction contracts featuring a total of 1450 apartments. Between 1989 and 1992, Steven operated a building design and documentation consultancy in Newcastle, primarily involved with redevelopment and repair work following the 1989 Newcastle earthquake. Steven's other interests include surfing, competitive cycling and naturism.

The correct citation for this article is:
Steven Fleming "Louis Kahn's Platonic Approach to Number and Geometry", pp. 95-107 in Nexus IV: Architecture and Mathematics, eds. Kim Williams and Jose Francisco Rodrigues, Fucecchio (Florence): Kim Williams Books, 2002.

Marie-Thérèse Zenner
Rue des Caves
41800 Fontaine-les-Côteaux FRANCE

From a technological point of view, the nave of Saint-Etienne in Nevers is recognized as the earliest known example of a triple elevation beneath a high vault. Curiously, the church features minimal use of traditional buttressing, relying instead on stilted quadrant arches and blind wall arcades. A standard structural analysis reveals that the quadrant arches function as interior "flying buttresses." This is the earliest known Romanesque example and amply predates Gothic developments in the use of perpendicular counterthrust. Most surprisingly, from its apex to the ground, the vaults and support systems closely describe a parabolic arc.

N2002-ZennerTechnical drawing of arcs (Paris, B.n., ms. fr 19093, fol. 21) with horse's head (fol. 18v) and calculation of wall thickness. Drawing by Renaud Beffeyte

A long-term study of this eleventh-century French church suggests the builders had access to a higher level of mathematical knowledge than previously admitted for the period. Design analysis, based on a measured survey, reveals probable use of three linear measures to determine both the plan and elevation. In contrast to Gothic design methods using sequential rectangles, this Romanesque design can be laid out using just three measures to describe a series of intersecting arcs (or circles). While these measures have a certain arithmetic relationship, a geometric relationship such as the golden section is not immediately obvious.

The three measures were crucial for planning the interior length and width of the plan, the principal vault heights and, in particular, the thickness of walls. The builder's concern to define mural mass at the design stage implies an interest in the stability of three-dimensional solid structures. Since the resolution of dynamic forces at Nevers was unprecedented, we propose that the choice of distance intervals was a key factor. This paper proposes to study the geometrical relationships of the three measures, in terms of intersecting curves and straight lines, within the context of the history of mathematics. We propose to examine whether the determination of such measures was intended for predicting stable three-dimensional solid structures.

The drawings accompanying this article were drawn on paper by hand using only compass and straightedge. The method qualifies as "constructive geometry," that is moving from point to point, and seems to closely approximate medieval building geometry. Subsequent analysis (by mathematical calculation or by means of AutoCAD) has shown that the angles of certain drawings are not regular. For example, the apex of the triangle in fig. 3 is 105°, not 108°. But the medieval mason did not use algebraic computation. And the ability to measure or draw precisely with a total station and AutoCAD is not necessarily appropriate to the medieval context -- which point should be measured where walls lean and axes deviate? On the general subjectivity of modern technology see Harrison Eiteljorg, II, "Measuring with Precision and Accuracy", CSA Newsletter, vol. XV, no. 1 (Spring 2002). Although it may not be "perfect," the proposed mnemonic device may nonetheless have existed as such and served as the basis for determining three proportions of "harmonic" relation; further research may tell us more. To take a second example, the pentagram in fig. 9 has angles of 34°, 37° and 35°, not 36°. Again while not absolute in the modern computational sense, the geometry of fig. 9 based on the golden section rectangle nonetheless did exist as such -- the sheep device was the "passport" of the medieval mason -- according to the continuous oral tradition. Although it may not land someone on the moon, this geometry sufficed as a mnemonic device for construction. Moreover, the organic quality of medieval building may be in part due to this built-in kind of irregularity.

My discussion of tiers point (figs. 4, 8) may be ignored, however, since the deviation is well more than 1%. Finally, the argument needs to take into account the chronology of highly-developed siege machines such as the trebuchet. It is generally assumed that the trebuchet was introduced in Europe only after the 1099 crusade but there is some suggestion that after moving westwards from China, beginning in the sixth century, knowledge of the machine had reached Byzantium and Sicily by the ninth century. See recent summary with references to Villard, in Peter Vemming Hansen, "Experimental Reconstruction of a Medieval Trebuchet," Acta Archaeologica 63 (1992): 189-268.

This email address is being protected from spambots. You need JavaScript enabled to view it. is an American scholar living and working in France. Recipient of numerous dissertation grants including the Fulbright and Whiting, she received the Ph.D. from Bryn Mawr College (Pennsylvania) in 1994 for a methodological study on the comprehensive physical analysis of medieval stone buildings. Associated with the CNRS in Poitiers (France) from 1994-2001, Dr. Zenner currently works as an independent historical consultant for museums and cultural sites in Europe (with her own agency, Villard Arts Inc). A new contributing editor for medieval studies to Nexus Network Journal, she also directs the annual conference program for AVISTA, an international interdisciplinary society she co-founded in 1984. Dr. Zenner is editor-in-chief of a volume in memory of Jean Gimpel (author of Les Bâtisseurs de Cathédrales), AVISTA series, volume 2, in preparation for Ashgate Publishing (England). Following a J. Paul Getty Postdoctoral Fellowship in 1996-97 entitled "The Sciences of Measure in Romanesque France between Vitruvius and Villard," Dr. Zenner has specialized in the mathematics as well as the monumental archaeology of pre-Gothic architecture, and more generally, the interrelationships of pre-modern science, technology and architecture, in addition to the symbolic traditions of astronomic-agronomic calendars in the pre-Christian and Christian eras.

The correct citation for this article is:
Marie-Thérèse Zenner, "Structural Stability and the Mathematics of Motion in Medieval Architecture", pp. 63-79 in Nexus IV: Architecture and Mathematics, eds. Kim Williams and Jose Francisco Rodrigues, Fucecchio (Florence): Kim Williams Books, 2002.

Alberto Pérez-Gómez
Director, History & Theory Graduate Program
School of Architecture
McGill University
815 Sherbrooke Street West
Montreal, PQ, CANADA H3A 2K6

Advocates of computer generated processes in architecture, all based on algorithmic languages, usually claim that these applications allow designers to bypass the questions of cultural specificity and ground design in scientific or natural principles, finally "closing the distance" between theory and practice. This is the right concern, but built upon wrong assumptions. The artistic imagination, issued from Romanticism, can be indeed dangerous or irrelevant. It should be condemned when it is abused as a vehicle for power, oppression and exploitation. Respect for the other and political correctness, however, is hardly assured by instrumental processes that operate in a historical and cultural vacuum. Only by engaging our own imagination (with its inescapable horizon of language, and despite its dangers) can we be truly compassionate. It is our imaginative faculty that allows us to identify with the other, and truly understand her suffering. This entails a very real, yet opaque connection between words and deeds.

Following this line of reasoning, I have argued in my work on representation for valorizing architectural work as process, rather than as a neutral means towards an end, driven by technological values. As embodied making, architecture is not only a means of formal discovery, it is also a vehicle for ethical production. This form of relationship between theory and practice, between words and process, is obviously not unprecedented in art, but has traditionally been less prevalent in architecture during the transformation of Western culture into modernity.

In my written contribution to this conference I offer two examples, from the fifteenth and twentieth centuries, that illustrate this relationship. The examples, the theories of Luca Pacioli and Le Corbusier, are intentionally far apart chronologically, and my use of them is unusual in contemporary scholarship. My aim is to draw a map of the vicissitudes of architecture as verb during the modernization of Europe, issuing not from a simplistic Platonic application, but from a realization of the affinity of architecture with Aristotle's middle sciences. Both theoreticians were interested in the Golden Section, geometry and mathematics, yet never as prescriptive tools, but rather as a discovery, accompanied by the self-consciousness of the "autonomy" of geometry as being ultimately "not of this world."

This email address is being protected from spambots. You need JavaScript enabled to view it. obtained his undergraduate degree in architecture and engineering in Mexico City, did postgraduate work at Cornell University, and was awarded a Master of Arts and a Ph.D. by the University of Essex in England. He has taught at universities in Mexico City, Houston, Syracuse, and Toronto, at the Architectural Association in London, and was Director of the Carleton University School of Architecture from 1983 to 1986. He has lectured extensively in North America and Europe. In January 1987 Pérez-Gómez was appointed Saidye Rosner Bronfman Professor of the History of Architecture at McGill University, where he is currently Director of Post-Professional (Master's and Doctoral) Programs, and chairs the History and Theory of Architecture division. From March 1990 to June 1993, he was also the Director of the Institut de recherche en histoire de l'architecture, a research institute co-sponsored by the Canadian Centre for Architecture, the Université de Montréal and McGill University.
His numerous articles have been published in the Journal of Architectural Education, AA Files, Arquitecturas Bis, Section A, VIA, Architectural Design, ARQ, SKALA, A+U, Perspecta, and many other periodicals. He is the author of Architecture and the Crisis of Modern Science (MIT Press, 1983; winner of the Alice Davis Hitchcock Award in 1984), Polyphilo or The Dark Forest Revisited (MIT Press, 1992), and together with co-author Louise Pelletier, Architectural Representation and the Perspective Hinge (MIT Press, 1997). He is co-editor of the book series entitled CHORA: Intervals in the Philosophy of Architecture (McGill-Queen's University Press) At present, Dr. Pérez-Gómez is engaged in a project to redefine the very nature of architectural education by revisiting its historical sources during the Enlightenment and the early nineteenth century, an urgent task after the failure of globalization which has become patent after September 2001.

The correct citation for this article is:
Alberto Pérez-Gómez, "Architecture as Verb and the Ethics of Making", pp. 35-46 in Nexus IV: Architecture and Mathematics, eds. Kim Williams and Jose Francisco Rodrigues, Fucecchio (Florence): Kim Williams Books, 2002.

Anthony Scibilia
University of Illinois
2, Avenue de Paris
78000 Versailles FRANCE

The geometric figure of similar triangles fundamental to the theorization and execution of images according to a system of one-point perspective in fifteenth-century Tuscany appears in a vast majority of later medieval basilican churches in Europe, beginning as early as the eleventh century. This figure appears in plan, but is palpable to the ambulatory observer in a series of axial alignments of pier/column edges located near the observer with pier/column/wall respond bodies further away. Though such alignments may be found in pre-eleventh-century architecture in Europe and beyond, it is only from the eleventh century on in Europe that these similar triangles are consistently lodged in the building's enclosing walls, a trait that links the perspectival spaces of such buildings with the phenomenon of the framed perspective painting and bas-relief, the triptych in particular.

N2002-ScibiliaSimilar triangles in the architecture of Brunelleschi:
a) towards a vanishing point ; b) towards an observer; c) towards an observer

The triptychal and perspective structure of these interiors is echoed in the overtly perspectival elaboration of the layered portal. As a radically compressed version of the church interior, the layered portal stands between the extensive and sprawling perspective space of architecture, and the collapse of such space into the flat painted panel.

The perspectival traits of later medieval basilican church interiors and portals differ throughout Europe, but two spatial types - one Tuscan, one northern - emerge, and display striking correspondence with the spatial and temporal structure of later Tuscan and northern painted perspective.

This email address is being protected from spambots. You need JavaScript enabled to view it. is a Ph.D. candidate in the Department of Art History and Archaeology at Columbia University in the City of New York, where he has earned M.A. and M. Phil. degrees, and taught for several years. In a doctoral dissertation titled "Perspective Before Perspective: The Spaces of Medieval Architecture," Mr. Scibilia shows that the spatial properties of fifteenth century perspective are anticipated in the design of medieval basilican churches dating from as early as the 11th century. Mr. Scibilia, who loves teaching, began to lecture as an undergraduate; prior to entering graduate school he had already spoken at several universities, colleges and seminars in the United States and abroad on the music and acoustics of Romanesque and Gothic churches, a thesis topic that earned him honors at Cornell University. As an architectural photographer, Mr. Scibilia has travelled extensively throughout Europe and the United States, and produced more than 10,000 images. His photographs have been published widely, and his work may be found in the libraries of many of the major universities, colleges, and museums in the United States. While still in high school, Mr. Scibilia was accepted to the Eastman School of Music, where he studied piano. After leaving Columbia, he plans to return to the piano, and attend the New England Conservatory of Music, where he has been invited by pianist Russell Sherman to teach on topics of visual art, music, and poetry.

The correct citation for this article is:
Anthony Scibilia, "Perspective Before Perspective", pp. 223-235 in Nexus IV: Architecture and Mathematics, eds. Kim Williams and Jose Francisco Rodrigues, Fucecchio (Florence): Kim Williams Books, 2002.